Difference between revisions of "Standard deviation"

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'''Standard deviation''' is a measure in [[Statistics|statistics]] for how much a set of values ''varies''. It allows for one to find how likely it is for a specific value to be obtained by doing a [[Z-test]].
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'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance]] of these values, where variance is defined as
  
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==Standard Deviation==
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:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>
  
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The standard deviation of a set of values is a measure of how widely the values differ from each other. Specifically, standard deviation follows the equation:
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where the [[expectation|expected value]] of ''X'' is E(''X'').
  
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:<math>\sigma(x) = \sqrt {\sum(x - \bar x) \over n - 1}</math>
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Thus the standard deviation is
  
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This is the square root of the variance, which is:
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:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>
  
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:<math>Var(x) = {\sum (x - \bar x) \over n - 1}</math>
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The formula for standard deviation must not be confused with the formula
  
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Where:
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:<math>S_{n} = \sqrt {\sum(x - \bar x) \over n - 1}</math>
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*<math>\bar x</math> is the arithmetic mean of all values of x
 
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*<math>\sum</math> is the [[summation]] function
 
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*<math>n</math> is the number of <math>x</math> values
 
  
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which is the formula for an [[estimator]] of the true standard deviation from a sample size of ''n''.  As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.
  
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If the distribution of the values is [[normal distribution|normal]] then it follows the [[Empirical rule]], which states that:
 
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:{{main|Empirical rule}}
 
  
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*68% of the values will fall within 1<math>\sigma</math> of the mean.
 
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*95% of all values will fall within 2<math>\sigma</math> of the mean.
 
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*99.7% of all values will fall within 3<math>\sigma</math> of the mean.
 
  
 
[[category:statistics]]
 
[[category:statistics]]

Revision as of 00:18, January 19, 2009

<math>\frac{d}{dx} \sin x=?\,</math> This article/section deals with mathematical concepts appropriate for late high school or early college.

Standard deviation is a measure in statistics of the dispersion of a set of values (represented as <math>X</math>). It is defined as the square root of the variance of these values, where variance is defined as

<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the expected value of X is E(X).

Thus the standard deviation is

<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

<math>S_{n} = \sqrt {\sum(x - \bar x) \over n - 1}</math>

which is the formula for an estimator of the true standard deviation from a sample size of n. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

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